Analytic geometry in a space

1003164406

Level: 
A
Determine whether any of the lines \( p \), \( q \) or \( r \) defined by the parametric equations given below passes through the coordinate origin. \begin{align*} p\colon x&=-2+4t, & q\colon x&=-5-5s, & r\colon x&=3-6u, \\ y&=1-2t, & y&=2-2s, & y&=-\frac12+u, \\ z&=-3+3t;\ t\in\mathbb R & z&=5+5s;\ s\in \mathbb R & z&=2-4u;\ u\in \mathbb R \end{align*}
Yes, it's the straight line \( r \).
Yes, it's the straight line \( p \).
Yes, it's the straight line \( q \).
None of the lines passes through the coordinate origin.

1003164405

Level: 
A
Determine whether the line \( p \) defined by parametric equations: \begin{align*} x&=-2+2t, \\ y&=1+3t, \\ z&=-3+3t;\ t\in\mathbb{R} \end{align*} intersects any of the coordinate axis.
Yes, it intersects the \( y \)-axis.
Yes, it intersects the \( x \)-axis.
Yes, it intersects the \( z \)-axis.
It intersects no coordinate axis.

1003164404

Level: 
A
Let a straight line \( p \) be defined by parametric equations: \begin{align*} x&=3+t, \\ y&=2-t, \\ z&=4;\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( xy \)-coordinate plane.
There is no such point \( M \).
\( M=[0;0;4] \)
\( M=[-3;2;0] \)
\( M=[1;-1;0] \)

1003164403

Level: 
A
Let a straight line $p$ be defined by parametric equations: \begin{align*} x&=-1+t, \\ y&=2+3t, \\ z&=5-t;\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( yz \)-coordinate plane.
\( M=[0;5;4] \)
\( M=[-1;0;0] \)
\( M=[0;3;-1] \)
\( M=[1;0;0] \)

1003164402

Level: 
A
Let a straight line \( p \) be defined by parametric equations: \begin{align*} x&=-1+2t, \\ y&=2+t, \\ z&=5-t;\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( xz \)-coordinate plane.
\( M=[-5;0;7] \)
\( M=[0;2;0] \)
\( M=[-1;0;5] \)
\( M=[2;0;-1] \)

1003164401

Level: 
A
Let a straight line \( p \) be defined by parametric equations: \begin{align*} x&=-1+2t, \\ y&=2+t, \\ z&=5-t;\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( xy \)-coordinate plane.
\( M=[9;7;0] \)
\( M=[0;0;5] \)
\( M=[-1;2;0] \)
\( M=[0;0;-1] \)

9000117404

Level: 
A
Determine whether the following planes are parallel, identical or intersecting. \[\begin{aligned} \rho \colon \frac{3} {8}x + \frac{1} {2}y -\frac{2} {3}z - 1 = 0,\qquad \sigma \colon \frac{3} {4}x + y -\frac{4} {3}z - 2 = 0 & & \end{aligned}\]
identical
parallel, not identical
intersecting

9000117406

Level: 
A
Determine whether the following planes are parallel, identical or intersecting. \[\begin{aligned} \rho \colon \frac{3} {2}x -\frac{1} {4}y + \frac{2} {3}z -\frac{2} {5} = 0,\qquad \sigma \colon \frac{2} {3}x - 4y + \frac{3} {2}z -\frac{5} {2} = 0 & & \end{aligned}\]
intersecting
identical
parallel, not identical